The resulting rift is known as the Great Schism (or, more specifically, the Great Western Schism, to distinguish it from the East–West Schism between the Roman and Orthodox Churches). Between 1378 and 1417, the Roman Church was divided between two—and, ultimately, three—competing papacies, each claiming to be legitimate and each denouncing the heresy of the others.
The Great Western Schism was a rift dividing the papacy, each claiming to be the legitimate power. I can't imagine having several different divisions of the papacy. Now I believe the Pope has absolute power in the church.